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If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in ametric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convexfunction, in the sense that it generates an ${\rm EVI}_0$-gradient flow, weprove that the pullback ${\sf E} \circ u : \Omega \to \mathbb{R}$ issubharmonic. This property was known in the smooth Riemannian manifold settingor with curvature restrictions on ${\rm X}$, while we prove it here in fullgenerality. In addition, we establish generalized maximum principles, in thesense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial \Omega$ controlsthe $L^p$ norm of ${\sf E} \circ u$ in $\Omega$ for some well-chosen exponents$p \geq q$, including the case $p=q=+\infty$. In particular, our results applywhen ${\sf E}$ is a geodesically convex entropy over the Wasserstein space, andthus settle some conjectures of Y. Brenier, "Extended Monge-Kantorovich theory"in Optimal transportation and applications (Martina Franca, 2001), volume 1813of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003.

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps