A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture

[This question is looking at the paper

• Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/cpa.3160310304, (pdf)]

My problem arises from (2.43) $$\Delta \varphi=f$$ where $$-m \leqq f \leqq C_{1} \exp \{C \sup \varphi\} \exp \left\{-\inf _{M} \varphi\right\}.$$

The paper already has a estimation of $$\sup\varphi$$ then the paper gets: $$\sup _{M}|\nabla \varphi| \leqq C_{6}\left(\exp \{-C \inf \varphi\}+\int_{M}|\varphi|\right)$$ How does this step use the Schauder estimate? Why is there a $$\int_{M}|\varphi|$$ on the right handside, should it be $$\sup|\varphi|$$?