The adjoint of a map

Question

Assume that $V=V(m,\mathbb{R})$, $W=W(n,\mathbb{R})$, and $g:V\to V^* $ and $G:W\to W^*$ are two isomorphisms. Given a map $f:V\to W$, define the adjoint of $f$, denoted by $\tilde{f}$, by $G(w,fv)=g(v,\tilde{f}w)$ (by Nakahar's book, page 98), where $v\in V$ and $w\in W$. Why does $\tilde{f}w$ exist so that $G(w,fv)=g(\tilde{f}w,v)$ and what is it exactly?

Answer 1

By the nice @Jakob comments and links, I answer my question in this way:

We define $\tilde{f}:W\to V$ by $\tilde{f}=g^{-1}\circ f^* \circ G$, where $f^*:W^* \to V^*$ is the pullback of $f$ (i.e. $f^* (g)=g\circ f$). Also we have $$G(w,fv)=G(w)(f(v))=f^*\circ G(w)(v)=g\circ \tilde{f}(w)(v)=g(\tilde{f}(w))(v)=g(\tilde{f}w,v).$$