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I've allready thought about this a bit but i don't want to reinvent the wheel. [Edit] So here's what i've been thinking of:

For every object $X\in\mathcal C$ we get a functor $[X,-]:\mathcal C\to\mathcal End(X)-\operatorname{mod}$. I think these functors are connected in a vaguely functorial way - hopefully by adjunctions between the module categories (adjunction in the 2-category of categories under $\mathcal C$).

My idea:

  • Let $f:X\to Y$ be a morphism in $\mathcal C_0$. On the one hand $[Y,X]$ is a bimodule from $\mathcal End(Y)$ to $\mathcal End(X)$. On the other hand $[Y,X]$ becomes a bimodule in the other direction - from $\mathcal End(X)$ $\mathcal End(Y)$ - by pre- and postcomposition with $f$ i.e. pulling back the module structure along $[f,f]:[Y,X]\to[X,Y]$. So assuming $\mathcal V$ is nice enough we have two functors $[Y,X]\otimes_{\mathcal End(X)}$ and $[Y,X]\otimes_{\mathcal End(Y)}$. However i can think of no canditate for a unit of a supposed adjunction between these two.
  • As in (2) $[Y,X]$ also becomes a semigroup object in $\mathcal V$ that has a (left/right) unit - and thus is a monoid - precisely when $f$ has an (left/right) inverse.

    • My vague guess is that the framework where this question could be handled is that of extranatural transformations.
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    Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ - it is actually a monoid in $\mathcal V$. Can the assignment $$X\mapsto \mathcal End(X)$$ be considered functorial in some way? Is there a language that captures the relations between $\mathcal End(X)$ when $X$ varies? What happens on the level of module categories $\mathcal V^{\mathcal End(X)}$?

    I've allready thought about this a bit but i don't want to reinvent the wheel.
    show/hide this revision's text 1

    Is assigning the endomorphism object in some sense functorial?

    Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's endomorphism object $\mathcal End(X):=[X,X]$ - actually a monoid in $\mathcal V$. Can the assignment $$X\mapsto \mathcal End(X)$$ be considered functorial in some way? Is there a language that captures the relations between $\mathcal End(X)$ when $X$ varies? What happens on the level of module categories $\mathcal V^{\mathcal End(X)}$?

    I've allready thought about this a bit but i don't want to reinvent the wheel.