Metric vs metrizable spaces A. Helemskii in the book "Lectures on functional analysis" write (in my horrible translation):

The category of Hausdorff topological spaces (morphisms are continuous maps) contain the full subcategory of metrizable topological spaces. Note: metrizable, not metric; the category of metric spaces (morphisms are continuous maps) is not a subcategory of the category of topological spaces (such as the category of linear spaces is not a subcategory of the category of sets).

Why the category of metric spaces $\neq$ the category of metrizable topological spaces?

A metric space is a pair $(X,d)$ consisting of a set $X$ and a metric $d$. In contrast, a metrizable space is topological space $(X,\tau)$ such that there is a metric $d$ which induces the topology $\tau$. This metric $d$ may not be unique.
There is a faithful functor $U:\mathbf{Met}\to\mathbf{MTop}$ which send $(X,d)$ to $(X,\tau)$ and a map $f$ to itself. Depending on what morphisms $\mathbf{Met}$ has, this may or may not be a full functor. If $U$ is not a full functor, then $\mathbf{Met}$ cannot be equivalent to $\mathbf{MTop}$.
On the other hand, if you take all the continuous maps as the morphisms in $\mathbf{Met}$, then $U$ is fully faithful. Since for each metrizable space $(X,\tau)$ there is a metric space $(X,d)$ with $U(X,d)=(X,\tau)$, $U$ is also surjective, which implies that it is part of an adjoint equivalence $(F,U;\text{Id}_{(X,\tau)},\varepsilon)$. As we see, the unit $\eta$ is the identity on the space $(X,\tau)$. For such adjunctions, the left adjoint $F$ is called a left-adjoint-right-inverse (The "-right-inverse" comes from the fact that if $\eta$ is the identity, then $UF$ is the identity functor, so $F$ is a right inverse to $U$.)
