An object $i$ is initial if for every $A$ there is a unique $i_A: i \to A$. The Dual notion is Terminal object. In case of sets and functions, an initial set is the empty set, and a terminal set is a one-element set.

unique isomorphism property: There is a unique isomorphism between any pair of initial objects; thus initial objects are ‘unique up to (unique) isomorphism’, and we can (and do) speak of the initial object (if any such exists).

For Vector spaces the zero-vector space is both initial and terminal.

Category Theory 4.1: Terminal and initial objects