In Lee's Introduction to Topological Manifolds (first edition) he states the following:
Suppose $A \subset X$ is a subspace. For any topological space $Y$, a map $f: Y \rightarrow A$ is continuous if and only if the following composite map from $Y$ to $X$ is continuous:
$$Y \stackrel{f}{\longrightarrow} A \stackrel{\iota_{A}}{\hookrightarrow} X$$
Proof. Directly from the definitions of continuity and the subspace topology,
$$f : Y \rightarrow A \text{ is continuous}$$
$$\iff \text{for all } U \underset{\text{open}}{\subset} A, f^{-1}(U) \underset{\text{open}}{\subset} Y $$
$$\iff \text{for all } V \underset{\text{open}}{\subset} X, f^{-1}(V \cap A) \underset{\text{open}}{\subset} Y $$
$$*\iff \text{for all } V \underset{\text{open}}{\subset} X, (\iota_{A} \circ f)^{-1}(V) \underset{\text{open}}{\subset} Y $$
$$\iff \iota_{A} \circ f : Y \rightarrow X \text{ is continuous} $$
I get everything before the *. How do I interpret $(\iota_{A} \circ f)^{-1}(V)$?
There are some other posts about the same proof or something similar here and here, but I am specifically interested in understanding the inverse step I just mentioned.
Thanks
