Is there a standard notation for the pre-composition operator?

Question

Let $X_1$, $X_2$, and $V$ be sets. Is there a standard name and a standard notation for the pre-composition operator $F$ that takes as input a function $\varphi:X_2^{X_1}$ and returns the operator $F_{\varphi}:V^{X_2}\rightarrow V^{X_1}$, which assigns for every function $f:V^{X_2}$ the function $F_{\varphi}(f)\in V^{X_1}$ defined by $$ F_{\varphi}(f) := f\circ\varphi $$

Is this what is meant by the term pullback in category theory? And, again, is there a standard notation for designating $F$, given $X_1$, $X_2$, and $V$?

Answer 1

In category theory, this would be denoted $\mathrm{Hom}_\mathbf{Set}(-,V)$, or simply $\mathbf{Set}(-,V)$. It's called the "contravariant hom-functor." That makes sense; given a function $$f : X_1 \rightarrow X_2,$$ we get a corresponding function $$\mathbf{Set}(f,V) : \mathbf{Set}(X_2,V) \rightarrow \mathbf{Set}(X_1,V)$$ going "the other way", defined by $$\mathbf{Set}(f,V)(g) = g \circ f.$$