nLab
endomorphism operad

Definition

For $(C,\otimes ,I)$ a monoidal category and $X\in C$ any object, the endomorphism operad ${\mathrm{End}}_C(X)$ of $X$ in $C$ is the operad whose set of $n$-ary operations is the hom-set

This comes with the obvious composition operation induced from the composition in $C$.

There are obvious generalizations of this definition to the enriched context and to the context of (∞,1)-operads.