Trying to follow the computation in https://arxiv.org/pdf/math/0602150.pdf, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \text{tr}_{g}h = g^{i \bar l} g^{k \bar j} R_{k \bar l} f^{\alpha}_{i} f^{\bar \beta}_{\bar j} h_{\alpha \beta} + g^{i \bar j} g^{k \bar l}f^{\alpha}_{i, k} f^{\bar \beta}_{\bar j, \bar l}h_{\alpha \bar \beta} - g^{i \bar j}g^{k \bar l} S_{\alpha \bar \beta \gamma \bar \delta }f^{\alpha}_{i}f^{\bar \beta}_{\bar j} f^{\gamma}_{k} f^{\bar \delta}_{\bar l}$.

The last term comes from the double derivative hitting on the metric $h$, but I am confused as to where does the extra $f^{\gamma}_{k} f^{\bar \delta}_{\bar l}$ come from.

Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they computed $\Delta \operatorname{tr}_{g}h = g^{i \bar l} g^{k \bar j} R_{k \bar l} f^{\alpha}_{i} f^{\bar \beta}_{\bar j} h_{\alpha \beta} + g^{i \bar j} g^{k \bar l}f^{\alpha}_{i, k} f^{\bar \beta}_{\bar j, \bar l}h_{\alpha \bar \beta} - g^{i \bar j}g^{k \bar l} S_{\alpha \bar \beta \gamma \bar \delta }f^{\alpha}_{i}f^{\bar \beta}_{\bar j} f^{\gamma}_{k} f^{\bar \delta}_{\bar l}$.

The last term comes from the double derivative hitting on the metric $h$, but I am confused as to where does the extra $f^{\gamma}_{k} f^{\bar \delta}_{\bar l}$ come from.