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. 



Algebra in a category
What you are talking about is the notion of monoid in a monoidal category. To show $A$ is a monoid ('algebra'), you need to construct a multiplication map $\mu: A \times A \to A$, that is associative, where $\times$ is the monoidal product for your monoidal category. In an example like a tensor algebra, you already have associativity, so what you need to show is that multiplication is a homomorphism of Yetter-Drinfeld modules.


Information on monoids in monoidal categories can be found on the nlab, or in the massive work of Aguiar and Mahajan on monoidal functors, species, and hopf algebras: <http://www.math.cornell.edu/~maguiar/a.pdf>.


