I am try to understand the concept: an algebra in a category. Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. $A$ is an algebra in $\mathcal{C}$ means the multiplication $m: A \otimes A \to A$ is a morphism in $\mathcal{C}$? 

Let $H$ be a bialgebra and $V$ a Yetter-Drinfeld module over $H$. Let $YD\_{H}^H$ be the monoidal category of all Yetter-Drinfeld modules over $H$. Since $YD\_H^H$ is a tensor category, the tensor algebra $T(V)$ is a Yetter-Drinfeld module over $H$. Therefore $T(V)$ is an object in $YD\_H^H$. When we want to show that $T(V)$ is an algebra in $YD\_H^H$, we need to show that the multiplication $m: T(V)\to T(V) \otimes T(V)$ is a homomorphsim of $H$-modules? That is, $h.(ab)=(h\_{(1)}.a)(h\_{(2)}.b)$ for $h \in H, a,b \in T(V)$. Is this correct? Thank you very much.