In set theory, suppose $A\subset B$. Then suppose we want to "replace $A$ by $C$". We have a notation for this: $B'=(B\setminus A) \cup C$.
However, suppose that $A,B,C$ are groups, or topological spaces, and $A$ is a subgroup or subspace of $B$, and that we want to generate a new group or topological space $B'$ by replacing $A$ by $C$. In this case, in order to remain faithful to the structure, we have to specify a particular way in which it's replaced, so we need extra data. Suppose we have a morphism $\phi:C\to A$ (homomorphism or continuous function), which tells us how the replacement is supposed to happen.
Is there a standard notation for denoting the equivalent of replacement $B'=(B\setminus A) \cup C$, in general categories other than Set? (I am not sure actually which assumptions about the category would be needed to define this).
Edit: I can imagine that something like $B[\phi:A\setminus C]$ is standard notation?