This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of monoids. Moreover, this extends to a result if $C$ has an enrichment in a monoidal category $\cal V$, and this changes the target to the category of $\cal V$-monoids.
My question is about the homotopical analogue of this. For example, if $C$ is a simplicial model category, Dwyer and Kan's sequence of papers on the simplicial localization show that there is a functor which captures the homotopy type of the monoid $End(X)$ properly and which is functorial in weak equivalences. (For a weak equivalence $X \to Y$, there are natural equivalences $End(X) \to Map(X,Y) \leftarrow End(Y)$, but these do not induce a map in the homotopy category of monoids.)
It is natural to ask this question in other circumstances. For instance, I was always taught that one of the operating principles of the theory of operads is that that algebra structures can be made invariant under weak equivalence. However, actually showing this seems to be surprisingly annoying, and requires one to make use of auxiliary constructions like the endomorphism operad of a map $f:X \to Y$. If this map is an acyclic cofibration or an acyclic fibration in an appropriate model category, this map has good homotopical properties. In general it seems to require constructing a factorization, and proving that we have a well-defined functor - from isomorphisms in the homotopy category of $C$ to the homotopy category of operads, or more generally some homotopy category of $\cal V$-monoids - seems to be ugly.
Is there a more direct proof that homotopy endomorphism monoids are functorial?
There are definitely obstructions to any direct constructions. For example, there is a quasi-isomorphism of chain complexes $$ (\mathbb Z \hookrightarrow \mathbb Z \oplus \mathbb Z) \to (\mathbb Z^2 \hookrightarrow \mathbb Z^2 \oplus \mathbb Z) $$ such that there is simply no map of differential graded algebras in either direction between the associated endomorphism objects.