Is a bijective morphism between metric spaces necessarily an isomorphism Does the inverse morphism for a bijective isometry necessarily preserve the metric or should the preservation of the metric for the inverse morphism be stated seperately? To make myself clear my question is that does the inverse morphism in metric spaces automatically preserve the distance (anologus to the case of algebraic structures that isomorphism is a bijective homomorphism) or is the situation like e.g. toplogical spaces that the continuity should be stated seperately for the invesre function in homeomorphisms?
EDIT: In short, the OP asks "If $f:X\to Y$ is a bijective map between metric spaces such that 
$$d_Y(f(x_1),f(x_2))=d_X(x_1,x_2) \; \; \; \forall x_1,x_2 \in X$$
then is the inverse map $f^{-1}:Y\to X$ also distance preserving i.e. do we have
$$d_X(f^{-1}(y_1),f^{-1}(y_2))=d_Y(y_1,y_2) \; \; \; \forall y_1,y_2 \in Y?$$

A metric space isomorphism is an isomorphism on the induced topologies is a bicontinuous bijection. It is possible to have a continuous bijection (at least in general topologies, not sure about metric spaces) that is not bicontinuous. 
