Tensors index raising and lowering as a mapping I am reading about tensor index raising and lowering done using the metric tensor and understood how it works geometrically for vectors and one-forms, i.e, I understand how a metric tensor $g_{\alpha \beta}$ is a map from a vector space to a vector space of one-form while lowering. 
$ V_{\alpha} = g_{\alpha \beta}V^{\beta} $
I am looking for a similar kind of understanding when we lower or raise indices of other higher order tensors using metric tensor. I am not able to explain what kind of map it is between say, $T^{\alpha \mu}_\gamma$ and $T^{\alpha}_{\beta \gamma}$ while lowering
$ T^{\alpha}_{\beta \gamma} =g_{\beta \mu} T^{\alpha \mu}_\gamma$  and how $g_{\beta \mu}$ is involved in the mapping. My text book says it's a map from second one-form argument of $T^{\alpha \mu}_\gamma$ to a vector, but I couldn't understand how this is the case as I am not even sure about domain and range of the map itself. Any help is appreciated. 
 A: Let $V$ be a finite dimensional vector space. If $g$ is non-degenerated symmetric $(0,2)$ tensor, then the map $\flat:V\rightarrow V^*$ given by formula
$$v^\flat(w)=g(v,w)$$
is an isomorphism. The inverse is dentoed by $\sharp:V^*\rightarrow V.$
If we have two linear maps (everything here is finite dimensional) $f_1:V_1\rightarrow W_1,f_2:V_2\rightarrow W_2$, then it inducess linear map $f_1\otimes f_2:V_1\otimes V_2\rightarrow W_1\otimes W_2$ given by
$$(f_1\otimes f_2)(v_1\otimes v_2)=f_1(v_1)\otimes f_2(v_2).$$
Moreover, if $f_1$ and $f_2$ are isomorphism, then $f_1\otimes f_2$ is an isomorphism as well.
Suppose we have a tensor of type $(2,1),$ so it is an element of $V\otimes V\otimes V^*.$ We wish to lower second and rise third index. So we consider maps (which are isomorphisms):
$$f_1:=id_V\hspace{10pt}f_2:=\flat\hspace{10pt}f_3:=\sharp$$
and define our isomorphism as
$$f_1\otimes f_2\otimes f_3=id_V\otimes\flat\otimes\sharp:V\otimes V\otimes V^*\rightarrow V\otimes V^*\otimes V.$$
Since we are dealing with isomorphisms we can "jump back and forth" between tensors (I mean lower and rise). 
I believe from that point you will figure out the general rule.
