We give a sufficient condition under which the moduli space of morphisms between logarithmic schemes is quasifinite over the moduli space of morphisms between the underlying schemes. This implies that the moduli space of stable maps from logarithmic curves to a target logarithmic scheme is finite over the moduli space of stable maps, and therefore that it has a projective coarse moduli space when the target is projective. The following corollary recovers and generalizes the results of Chen,

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... movich-Chen, and Gross-Siebert mentioned earlier. Corollary 1.2. Let Y be a fine, saturated logarithmic scheme over S. Fix a type u (see Definition 3.1). Let M u (Y ) be the moduli space of stable maps from logarithmic curves to Y of type u, let M u (Y ) be its underlying algebraic stack, and let M (Y ) be the moduli space of stable maps to Y . Then M u (Y ) is finite over M (Y ). Proof. It has been shown elsewhere that the moduli space of stable logarithmic maps is locally of finite type [Wis16, Theorem 1.1], bounded [ACMW17, Proposition 1.5.6], and satisfies the valuative criterion for properness [ACMW17, Proposition 1.4.3] over stable maps, so it remains only to show that the geometric fibers are finite. It is therefore sufficient to consider a base S whose underlying scheme is the spectrum of an algebraically closed field and fix a logarithmic curve X over S. This corresponds to a map from S to the moduli space of curves, and we wish to show that the base change M u,X (Y ) of M u (Y ) via this map is finite over the base change M u,X (Y ) of M u (Y ). The space M u,X (Y ) is the saturation 1 of the logarithmic algebraic space Hom LogSch/S (X, Y ) u obtained from Hom LogSch/S (X, Y ) by base change via u : S → T . Therefore, the projection from M u,X (Y ) to S can be factored: