Does category-theory have an interesting perspective on the phrase 'under the induced operations'? Category theory often gives context or perspective to phrases that are ubiquitous throughout mathematics. Now we often make statements like: "the set $X$ becomes a [whatever] under the induced operations." For example:


*

*Given an algebraic theory $T$ and a $T$-algebra $X$, the set of all functions $k \rightarrow U(X)$ forms an algebraic structure under the induced operations, where $k$ is a set and $U$ is notation for the underlying set functor. This turns out to satisfy the universal property of $X^k.$

*Given an Abelian group $X$, the set of all finitely-supported functions $k \rightarrow U(X)$ forms an Abelian group under the induced operations. It satisfies the universal property of `the coproduct of $X$ with itself $k$ many times.'

*Once again starting off with an Abelian group $X$, the set of endomorphisms from $X$ back to itself forms an Abelian group under the induced operations.

Question. Does category-theory have an interesting perspective on the phrase 'under the induced operations'?

 A: The question is way too broad. Therefore my answer is also broad. Let me only give some key notions which fit to the mentioned examples..


*

*Creation of limits

*Internal hom

*Topological category

*Grothendieck fibration
A: I agree that this question is way too broad. I think you are pointing to more than one phenomenon and I am not sure which one is of the most interest to you, so let me just say some things and you can tell me what you actually care about. 
For starters, let $G$ be a group object in a category $C$ with finite products, and let $X$ be an object. Then $\text{Hom}(X, G)$ naturally acquires the structure of a group "under the induced operations." Why? Because the Yoneda embedding $G \mapsto \text{Hom}(-, G)$ is continuous, and in particular preserves finite products, and functors preserving finite products send group objects to group objects. (And why is that? Because the definition of a group object just involves some equations between morphisms between finite products of an object. All functors preserve equations between morphisms, so the only remaining thing to preserve is products.) 
This generalizes in a few different directions. For example, "group objects" can be replaced with "models of a Lawvere theory" and the argument is exactly the same. It's also true that monoidal functors send monoid objects to monoid objects; this is one way of explaining why, for example, the free vector space functor sends monoids to algebras. 
Your third example is a little different. There the key observation is that the category of abelian groups is enriched over itself, so in fact if $A, B$ are two abelian groups then $\text{Hom}(A, B)$ is also naturally an abelian group. If we moreover look at endomorphisms then an additional thing happens: $\text{Hom}(A, A)$ is not only an abelian group, but in fact a "monoid enriched in abelian groups," which is just a very roundabout way of saying a monoid object in abelian groups, or a ring. 
There's another thing you might be pointing to which has something to do with why a set-theoretic construction ends up having not only extra structure but a universal property. In your first example it's just the observation that forgetful functors tend to preserve limits; for example, whenever a forgetful functor has a left adjoint, it always preserves limits. 
