Let $X$ be a smooth compact manifold. Let say I want to define a category of vector bundles over $X$, where the objects are of the form $(E, g^E, \nabla^E)$ ($E\to X$ is a complex vector bundle, $g^E$ is a Hermitian metric and $\nabla^E$ is a unitary connection), and the morphisms are $T:(E, g^E, \nabla^E)\to(F, g^F, \nabla^F)$, where $T:E\to F$ is a bundle isomorphism preserving the metrics, i.e. $g^E=T^*g^F$. So an isomorphism between two objects is simply a morphism.
It seems to be mathematically correct to define such a category (correct me if it's not), but it looks awfully strange. Do we have other (popular) examples for which isomorphisms are morphisms?