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Terminal Object**

An object \(\tau\) of a category \(\mathcal{C}\) is called 'terminal' if, for any other object \(x\) of \(\mathcal{C}\), there is a unique arrow from \(x\) to \(\tau\), i.e. \(\forall x\in\mathcal{C}_0:\exists! \tau_x\in\mathcal{C}_1:\mathrm{dom}(\tau_x)=x\land\mathrm{cod}(\tau_x)=\tau\) or (in terms of hom-sets) \(\forall x\in\mathcal{C}_0:\mathrm{Hom}_\mathcal{C}(x,\tau)=\{\tau_x\}\). If a category has terminal objects they are all isomorphic to each other.