Does the exponential map of a Lie Group satisfy a universal property?

Question

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $exp: \mathfrak{g} \to G$ be the exponential map. The map $exp$ gives a diffeomorphism of a neighborhood of the identity $0_{\mathfrak{g}}$ in $\mathfrak{g}$ with a neighborhood of the identity $1_{G}$ in $G$, and the linearization $exp_{*}: \mathfrak{g} \to \mathfrak{g}$ is the identity map up to an identification. Does $exp$ satisfy a Universal Property? In particular , does any map $f: \mathfrak{g} \to G$ satisfying the above properties factor as $g \circ exp$ for some smooth homomorphism $g: G \to G$? My apologies if this is a trivial question, I am still very much a beginner in Lie Theory.