There are multiple cases of what is described in the title, but here is the one which prompted my question:
There are two alternate definitions of a lattice. One is as a poset $(L,\geq)$ where meet and join exist for every pair of elements, and one is as an algebraic structure $(L,\land, \lor)$ with two binary operations satisfying certain requirements. These two definitions are equivalent in the following sense:
Let $C$ be the category of lattices as posets, and $D$ be the category of lattices as algebraic structures. Given an object $(L,\geq)$ of $C$, we can define the operations $\land$ and $\lor$ on $L$, which defines a map $F:Ob(C)\to Ob(D)$. Given an algebraic lattice $(L,\land, \lor)$ we can define an order $\geq$ on $L$, which gives us a map $G:Ob(D)\to Ob(C)$. These two maps are inverses of each other, so a lattice in either sense has all the information necessary to reconstruct a lattice in the other sense.
My question is: the maps $F$ and $G$ are in fact an equivalence of categories, and this is what makes the above claim that the definitions are "equivalent" work. But we haven't actually defined how $F$ and $G$ behave on morphisms. How can it be the case that we are defining a pair of functors without specifying their behavior on morphisms?
Actually, this is not precisely what I mean to ask. It's clear to me that their behavior on morphisms simply doesn't need to be stated, because it follows canonically from their definition on objects. My question is rather something like: why is it canonical? Where is the information coming from that allows us to infer the behavior on morphisms without stating it?
This question goes equally for other cases of category equivalence that are often stated without reference to morphisms, like the equivalence of the various definitions of a topological space.
You did not define functors at all. You defined two classes; the class of algebraic lattices, and the class of poset lattices. You then constructed a bijection between them.
At this point, all you have done is construct a bijection between the two classes. Only once you define morphisms between poset lattices and between algebraic lattices can you then extend this bijection to an isomorphism of categories.
There are at least two sensible notions of morphisms for a category whose objects are posets which are lattices. One is monotone maps, which makes the category of poset lattices a full subcategory of the category of posets. The other is the algebraic notion of lattice homomorphism - maps that preserve finite meets and joins. You would need to specify that you’re using the latter category to be maximally precise.