Automorphism of infinite graphs

Question

An automorphism on a finite graph $G$ is defined as an isomorphic map from $G$ to itself. How is automorphism defined on infinite graphs? For example, on the ray graph $G_r: 1 \rightarrow 2 \rightarrow 3 \rightarrow \cdots$, the identity map is obviously an automorphism. But what about the map $f(G_r) = 2 \rightarrow 3 \rightarrow 4 \rightarrow \cdots$? It is an isomorphism, and also maps $G_r$ “to itself” in the sense that the image is a subgraph of $G_r$.