Exact sequence of holomorphic vector bundles

Question

Let $A,B,C$ and $T$ be four holomorphic vector bundles on the complex manifold $X$. If I have a short exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ is it true that the sequence $$0\longrightarrow A\otimes T\longrightarrow B\otimes T\longrightarrow C\otimes T\longrightarrow 0$$ is also exact? I think that the answer is positive because by the shrt exact sequence on the fibers $$0\longrightarrow A_x\longrightarrow B_x\longrightarrow C_x\longrightarrow 0$$ we obtain a short exact sequence $$0\longrightarrow A_x\otimes T_x\longrightarrow B_x\otimes T_x\longrightarrow C_x\otimes T_x\longrightarrow 0.$$ I have only to prove that the the maps $ A\otimes T\longrightarrow B\otimes T, \: B\otimes T\longrightarrow C\otimes T$ are holomorphic. Any helps?