Proofs of uniqueness up to isomorphism I'm embarrassed about this question but here goes:
Reading through the proof of Lambek's lemma (https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor), I was stumped at the step where $\alpha \circ i$ is declared equal to ${1\!\!1}_{X} $. I stop at $\alpha \circ i : X \rightarrow X $.
Elsewhere, I am similarly stuck at proofs of uniqueness (up to isomorphism) of initial/terminal object (https://proofwiki.org/wiki/Initial_Object_is_Unique). Why is $v \circ u$ equal to ${1\!\!1}_{O} $? As far as I can see from the diagram, $v \circ u$ could be any arrow of type $X \rightarrow X $.
I feel I'm missing something basic here and I'm frustrated.
 A: For the initial object: there is only one arrow from an initial object to itself, as there is only one arrow from an initial object to any object. Since the initial object has the identity morphism to itself, each arrow from an initial object to itself must be equal to that arrow, the identity.
This generalizes to other objects with universal properties; typically these are phrased as "given an object with some arrows ... such that ... there exists a unique arrow ... such that ...". Then, in the case of the object itself, the unique arrow that the statement guarantees is the identity arrow, and this is how you prove such a statement about unique isomorphisms.
For initial objects, the above sentence becomes: "given an object $X$ with [no arrows] such that [no conditions], there exists a unique arrow $0 \to X$ such that [no conditions]." It's a somewhat trivial case.
Also, as a general hint: if you are just starting out with category theory, steer clear of the n-Lab if you can help it -- it's a useful resource, but not for beginners.
