When is the functor $\otimes:\mathcal{C}^\omega\times\mathcal{C}^\omega$ is monoidal functor? Admittedly, the question in the title is not a yet a precise question, so I must make it a precise question. Let $\mathcal{C}$ be a monoidal category, with monoidal product, $\otimes$. We will let $\omega$ be the set of finite ordinals regarded as a category where the order on the finite ordinals gives the needed homsets. The category, $\mathcal{C}^\omega$, then is the category of diagrams of the form, $$X_0\to X_1\to X_2\cdots,$$ and the morphisms are a sequence of maps, making the evident squares commute. Now let us define a category, which we will denote $P_n$. This will be the full subcategory of $\omega\times\omega$ such that every object, $(p,q)$ satisfies the condition that $p+q\leq n$. Now let $X,Y$ be two objects of $\mathcal{C}^\omega$. We define a functor which we will denote, $F_{X,Y,n}:P_n\to \mathcal{C}$ defined as $F_{X,Y,n}(i,j)=X_i\otimes Y_j$. If $i_1<i_2$, and $j_1<j_2$, then we have maps $\phi:X_{i_1}\to X_{i_2}$ and $\theta:Y_{j_1}\to Y_{j_2}$. Then $F_{X,Y,n}((i_1,i_2)\to (j_1,j_2))$ is the tensor product of the two maps $\phi\otimes\theta$. We then define the tensor product, $(X\otimes Y)_n$ to be $colim\mbox{  }F_{X,Y,n}$. The universal properties of the colimit allows us to show that we have canonical maps, $(X\otimes Y)_n\to (X\otimes Y)_{n+1}.$
My questions are as follows:

If $\mathcal{C}$ is a monoidal functor, (with only associativity and unit constraints) is the bifunctor defined on $\mathcal{C}^\omega$ also a monoidal functor? If not what are some necessary and sufficient conditions on $\mathcal{C}$ to make this happen?  

I am also interested in the same question if the monoidal product is braided, symmetric and a categorical product. 
Part of the motivation for this is that it generalizes the product on the category of $CW$ complexes and filtered spaces. I would like to see how far this construction may be generalized. Any references would be welcome.
 A: So you define a tensor product on $\omega$-sequences by $(X \otimes Y)_n = \varinjlim_{i+j \leq n} X_i \otimes Y_j$ and ask if this is a monoidal structure on the category of $\omega$-sequences.
Let us assume that the ambient monoidal category has (directed) colimits and that the tensor product preserves these colimits in each variable (note that this is not the case for $\mathsf{Top}$, we rather have to choose a convenient category of topological spaces such as $\mathsf{CGHaus}$). Then I think that the answer is Yes.
Define the unit by $1 := (1 \xrightarrow{\mathrm{id}} 1  \xrightarrow{\mathrm{id}} 1  \xrightarrow{\mathrm{id}} \dotsc )$. Then $(X \otimes 1)_n$ is the colimit of the diagram
\begin{array}{c} X_0 & \to & X_1 & \to & \dotsc & X_{n-1} &  \to & X_n \\
\downarrow & & \downarrow & & & \downarrow & &   \\
X_0 & \to & X_1 & \to & \dotsc & X_{n-1}  & \\
\downarrow & & \downarrow & & & &  \\
\vdots & & \vdots & & & &  \\
X_0 & \to & X_1 & \\
\downarrow &\\
X_0 & 
\end{array}
which is obviously $X_n$. This shows $X \otimes 1 \cong X$, clearly natural in $X$. A similar argument shows $1 \otimes X \cong X$. Now let $X,Y,Z$ be sequences. Then we have
$ (X \otimes (Y \otimes Z))_n =\varinjlim_{i+j \leq n} \bigl(X_i \otimes \varinjlim_{p+q \leq j} (Y_p \otimes Z_q)\bigr) =  \varinjlim_{i+j \leq n} \varinjlim_{p+q \leq j} X_i \otimes (Y_p \otimes Z_q)$
$= \varinjlim_{i+j \leq n,~~ p+q \leq j} X_i \otimes (Y_p \otimes Z_q) = \varinjlim_{i+p+q \leq n} X_i \otimes (Y_p \otimes Z_q)$
where in the last step we used a cofinality argument. Now we can use the natural isomorphisms $X_i \otimes (Y_p \otimes Z_q) \cong (X_i \otimes Y_p) \otimes Z_q$ to apply the same argument backwards to get an isomorphism $(X \otimes (Y \otimes Z))_n \cong ((X \otimes Y) \otimes Z)_n$. One checks easily that these isomorphisms are natural in $n$, so that $X \otimes (Y \otimes Z) \cong (X \otimes Y) \otimes Z$, which is also clearly natural in $X,Y,Z$. Now one has to work out that the coherence diagrams commute. But these are just colimits of the corresponding coherence diagrams in the given monoidal category. I won't spell out the details here. This construction also works for braided or symmetric monoidal categories.
More generally, if $N$ is a small monoidal category and $\mathcal{C}$ is a monoidal category with colimits which distribute over the tensor product, then the category of functors $C^N$ is again a monoidal category with the monoidal structure given by
$(X \otimes Y)_n = \varinjlim_{i \otimes j \to n} X_i \otimes Y_j.$
I think this is well-known, but unfortunately I don't know a reference. Perhaps someone can add a reference. It is kind of similar - but not identical - to Day convolution.
