Examples of categories which naturally include End(O) as object I want examples of categories $\textbf C$ which naturally include $End_{\textbf C}(O)$ as object for objects $O$ in the category. The set of all endomorphims is always a monoid under the composition of morphisms, but beside from that.
Counterexamples is also interesting.
I'm not asking for examples that has to be of interest from category theoretical point of view, but just natural examples where the structure of $O$ transfers to $End_{\textbf C}(O)$. The reason for my question is that I imagine some kind of correspondence between those categories and categories with tensor products, since there is a diagram that recalls about the universal property of tensor products:
$\require{AMScd}$
\begin{CD}
    X\times Y @>\mu>> End_\textbf C(X\times Y)\\
    @V \forall h V V\# @VV \exists\varphi V\\
    Z @= Z
    \end{CD}
$\mu(x,y)(a,b)=(x,y)$, $\varphi(f)=h\psi_{a,b}$ and $\psi_{a,b}(f)=f(a,b)$.
$\mu$ and $h$ are bi-morphisms and $\varphi$ is a morphism.

Examples:


*

*Set, $End(X)$ is trivially a set.

*$R$-Mod, $End(X)$ is a $R$-module:
$(\alpha+\beta)(x)=\alpha(x)+\beta(x)$; 
$(r\cdot\alpha)(x)=r\cdot\alpha(x)$ for commutative rings. For non commutative rings $End(X)$ is only a $\mathbb Z$-module.

*Grp, no obvious group structure on $End(X)$.

*Top, $End(X)$ is a topological space with the topology of pointwise convergence.


$End(O)$ is a set (or a monoid), it belongs to the category of sets and nothing else.
Perhaps you are interested in the notion of a cartesian closed category. There one has an internal hom object $\underline{\hom}(x,y)$ for all objects $x,y$, in particular $\underline{\mathrm{End}}(x):=\underline{\hom}(x,x)$.
