Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks and this limits the utility of span categories as a formal framework. If $\mathcal{C}$ and $\mathcal{C} ′$ are categories and $F$ is a functor from $\mathcal{C}$ to $\mathcal{C} ′$, we introduce the notion of an $F$-pullback of a cospan in $\mathcal{C}$ as well as the notion of span tightness of $F$. If $F$ is span tight, then we can form a generalized span category $Span(\mathcal{C} , F)$, which allows us to find a way around the technical difficulty of $\mathcal{C}$ failing to have pullbacks.