If the lectern before us were made of plastic instead of wood, it would be a different lectern to the wooden one that is actually before us. — Banno
Being made of wood is not essential of being a lectern.............By the law of identity all properties of a particular individual are essential properties.......... So we cannot say that it is necessarily made of wood, that might be a mistaken judgement. — Metaphysician Undercover
The law of identity
There is something in front of me. It has many properties: being made of wood, brown in colour, being in a lecture room, being 1.5m in height, not made of ice, etc. It must be true as @Metaphysician says: "By the law of identity all properties of a particular individual are essential properties". In the sense that all the properties this thing has are essential to making this thing, in that if this thing had different properties it would be a different thing. If this thing lost even one molecule, it would be a different thing. It must be true as @Banno wrote: "If the lectern before us were made of plastic instead of wood, it would be a different lectern to the wooden one that is actually before us."
We judge some properties of an object more essential than others
However, this would be inconvenient for humans in navigating their world if everything they saw in the world was continually changing. Therefore, for convenience, humans judge certain properties more essential than others. For example, one person could judge that being made of wood was more essential to being a lectern, and another person could judge that being in a lecture room was more essential to being a lectern. There is no correct judgement, it is a matter of personal judgement. It must be true as @Metaphysician wrote: "Being made of wood is not essential of being a lectern."
Kripke and Rigid Desgnators
Kripke wrote: i) "What do I mean by ‘rigid designator’? I mean a term that designates the same object in all possible worlds.", ii) "We can talk about this very object, and whether it could have had certain properties which it does not in fact have. For example, it could have been in another room from the room it in fact is in, even at this very time, but it could not have been made from the very beginning from water frozen into ice."
In order to make sense of objects in the world, those properties judged more essential than others required to maintain the identity of an object, are called by Kripke "rigid designators". Rigid designators are defined by personal judgement. For example, let a rigid designator of a lectern be "being made of wood", though it could equally well have been "being in a lecture room". As we have judged being made of wood is an essential property of a lectern, by definition, being made of wood becomes a property of a lectern.
It follows that as "being made of wood" is now part of the definition of a lectern. If I see an object that is not made of wood, then by definition it is not a lectern. It must be true as @Metaphysician wrote: "So we cannot say that it is necessarily made of wood, that might be a mistaken judgement". Because this definition of lectern doesn't include being in a lecture room, as a lectern may or may not be in a lecture room, the lectern is not necessarily in a lecture room.
Definitions are necessarily true a priori
Kripke wrote: i) "So we have to say that though we cannot know a priori whether this table was made of ice or not, given that it is not made of ice, it is necessarily not made of ice. " ii) "For example, being made of wood, and not of ice, might be an essential property of this lectern."
But what is a "lectern". What does "lectern" mean. The meaning of words cannot come empirically from observation of the world, in that, if I look at the world and see on the one hand a group of charging elephants and on the other hand a stand made of wood in a lecture room, it would be impossible to know a posteriori which of these "lectern" refers to. The meaning of "lectern" can only be determined a priori either from a dictionary or similar or from use within language. Therefore, the meaning of "lectern" is a priori and necessary.
Therefore, in Kripke's statement "being made of wood..........might be an essential property of this lectern.", "this lectern" may be replaced by "this something that is known a priori as being made of wood". This gives the statement "being made of wood..........might be an essential property of this something that is known a priori as being made of wood", which is an analytic statement, and from the law of identity, being made of wood is necessarily being made of wood.
Conclusion
Therefore, as a lectern has been defined as being made of wood, if this lectern is made of wood, then this lectern is necessarily made of wood. Such identity statements are therefore necessary and a priori.