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The category can be drastically different. For example, suppose your morphisms are precisely local homeomorphisms. Call this category $Top^{et}$. This category is locally a topos, which is something certainly not the case for the category $Top$ of all continuous maps. Moreover, $Top^{et}$ lacks a terminal object (which is something which will happen for many variants). In fact, $Top^{et}$ behaves much more like a category of sheaves on a single space, then a category of spaces; to see this, if we let $\mathfrak{Top}^{et}$ denote the bicategory of (etale) topological stacks and local homeomorphisms, this is equivalent to the bicategory of stacks on some (filtered colimit of) etale topological stack(s). The bicategory $\mathfrak{Top}^{et}$ contains $Top^{et}$ as a full subcategory, and (if we restrict to a set of topological spaces) is a $2$-topos, so has all the limits, colimits etc. you can imagine. Using this, it can be shown that $Top^{et}$ does have at least binary products, but they behave very differently than in $Top.$ For example, instead of $Top$, consider the category $Mfd$ of smooth manifolds. Given an $n$-manifold $N$ and an $m$-manifold $M,$ their product $N \times^{et} M$ in $Mfd^{et}$ is empty if $n \ne m,$ and if $n=m,$ their product is a highly non-Hausdorff smooth $n$-manifold. This is discussed in http://arxiv.org/abs/1212.2282.