I'm trying to understand the stucture regular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_2(\mathbb{C})$, given in part iii) of Theorem 5.3.1 in Mazorchuk's Lectures on $sl_2(\mathbb{C})$-modules. The notation of the situation is very specific, so I've tried to write it out the part that is confusing me more simply.
Suppose you have a category with two simple objects, $A$ and $B$. Let $P(A)$ and $P(B)$ be their indecomposable projective covers. It is known that $P(B)$ has composition factors $B,A$, and $P(A)$ has composition factors $A,B,A$ (written in "descending" order, with multiplicity).
The text goes on to say... Since $A\subset P(B)$, and occurs one as a subquotient of $P(B)$, and $P(A)$ is projective with simple top $A$, we have a unique, up to a nonzero scalar, nonzero morphism $\varphi\colon P(A)\to P(B)$. As $B$ occurs with multiplicity one as a subquotient of $P(A)$, and $P(B)$ is projective, we have a unique, up to nonzero scalar, nonzero morphism $\psi\colon P(B)\to P(A)$.
I don't get why there only exist unique nonzero maps up to a scalar, and how it relates to the uniqueness of a composition factor.
This is a general thing for suitably nice categories like this (i.e. linear categories over an algebraically closed field, which has finite length, finite dimensional Hom-spaces, and where the simple objects have projective covers).
In this sort of category, the dimension of $\operatorname{Hom}(P(A),M)$ is the multiplicity of $A$ in $M$, for any object $M$.
(I assume the word "isomorphism" near the end is a typo for "morphism").