Left Kan extension of a $\mathsf{Set}$-valued finite-product-preserving functor I've been told that the following is true:

Proposition. Consider $\mathcal A,\mathcal B$ small categories with finite products and $j\colon \mathcal A \to \mathcal B$ preserving them. Then for any finite-product-preserving $F \colon \mathcal A \to \mathsf{Set}$, its left Kan extension $j_!F$ also is finite-product-preserving.

And I'm inclined to believe it because of the case where $\mathcal A = \mathsf{FinSet}$ and $j$ is bijective-on-objects: it gives that the forgetful functor for the Lawvere theory $j$ admits a left adjoint (the free functor). 
Here's my attemp at proving the proposition:


*

*Remark that for any $a \in \mathcal A$, the left Kan extension $j_!{\mathcal A(a,-)}$ is just $\mathcal B(j(a),-)$.

*For $F \colon \mathcal A \to \mathsf{Set}$ considered as a preasheaf on $\mathcal A^\circ$ (opposite category of $\mathcal A$), one has
$$ F = \operatorname{colim}\left( \mathcal A^\circ / F \to \mathcal A^\circ \stackrel h \to [\mathcal A,\mathsf{Set}] \right) $$ 
where $h$ is the Yoneda embedding.

*Left adjoints commute with colimits, so
$$j_!F = \operatorname{colim}_{f:h(a)\to F}\left( \mathcal B(j(a),-) \right) $$ 

*Functors of the form $\mathcal B(b,-)$ preserves finite products by definition, and filtered colimits commute with finite limits in $\mathsf{Set}$. So we conclude by showing that $\mathcal A^\circ / F$ is filtered.

*As $F$ preserves binary products, any diagram over the discrete category with two objects admits a cocone. 

But what about the diagram over $\bullet \rightrightarrows \bullet$ ?

I don't see why those should admit a cocone... I didn't yet take advantage of the fact that the terminal of $\mathcal A$ is mapped by $F$ to a singleton, but it does not seem to help. Also, I did not use the finite-product-preserving hypothesis on $j$, which I believe is necessary.
Can someone help me finish/correct this sketch of proof ? (Or provide a counter-example if the proposition is false.)
 A: You can prove the statement without appealing to the sifted VS discrete commutation between limits and colimits; it's a far more general result about promonoidal functors.
This result tells you that if you are given a strong promonoidal functor $j\colon \mathcal{A}\to \mathcal{B}$  between promonoidal categories, then the left Kan extension $\text{Lan}_j\colon [\mathcal{A},\mathbf{Set}]\to [\mathcal{B},\mathbf{Set}]$ is a strong monoidal functor, when the presheaf-categories are endowed with the convolution product
$$
F\star G = \int^{A', A''} \mathbf{P}(A', A'',-)\times FA' \times GA''. 
$$
This boils down to your statement choosing the "trivial" promonoidal structure $\mathbf{P}(A', A'', X) = \hom(A', X)\times \hom(A'',X)$, as in that case $(F\star G)(X)\cong FX \times GX$.[1]
A promonoidal functor between promonoidal categories $h : (\mathcal A, P_{\cal A}, J_{\cal A})\to (\mathcal B, P_{\cal B}, J_{\cal B})$ is a functor $h\colon \mathcal A\to \mathcal B$ endowed with natural transformations
$$
\begin{gather}
P_{\cal A}(A,A'; A'') \to P_{\cal B}(hA,hA'; hA'') \\
J_{\cal A}A \to J_{\cal B}(hA)
\end{gather}
$$
which induce natural isomorphisms
$$
\begin{gather}
\int^{A''}P_{\cal A}(A, A';A'')\times \hom(hA'', B)\cong P_{\cal B}(hA, hA'; B) \\
\int^A J_{\cal A}A \times \hom(hA, B)\cong J_{\cal B}B
\end{gather}
$$
Here's the coend rollercoaster (the trick is that $\text{Lan}_UV$ is the coend $\int^X \hom(UX,-)\times VX$)[2]:
$$\begin{align} 
\text{Lan}_j(F\star G)(B) &\cong \int^A \mathcal{B}^{jA}_B \times (F\star G)_A\\
(1)&\cong \int^A \mathcal{B}^{jA}_B \times \left[ \int^{A', A''}P_A^{A'A''}\times F_{A'}\times G_{A''}\right]\\
(2)&\cong \int^{AA'A''}\mathcal{B}^{jA}_B \times P_A^{A'A''}\times F_{A'}\times G_{A''}\\
(3)&\cong \int^{A'A''}  P_B^{jA'jA''}\times F_{A'}\times G_{A''}\\
(4)&\cong \int^{A'A''B'B''}\mathcal{B}^{jA'}_{B'}\times \mathcal{B}^{jA''}_{B''}\times P_B^{B',B''}\times F_{A'}\times G_{A''}\\
(5)&\cong \int^{B'B''}P(B',B'',B)\times \left[\int^{A'} \mathcal{B}^{jA'}_{B'}\times F_{A'}\right ]\times \left[\int^{A''} \mathcal{B}^{jA''}_{B''}\times G_{A''}\right ]\\
&\cong (\text{Lan}_j F\star \text{Lan}_j G)(B)
\end{align}$$
(1) is true by definition; (2) is true since $\bf Set$ is cartesian closed and products distribute over colimits; (3) is true since the functor $j$ is strong promonoidal; (4) is true, it's a form of Yoneda expansion/reduction; to obtain (5) you only have to reorder factors.

[1] Do not worry, I'm not assuming you're comfortable with the definition of promonoidal category; the idea behind them is that a promonoidal category is what you obtain when you take the definition of a
monoidal category and you replace every occurrence of the word functor with the word profunctor. Appendix A here contains a basic introduction to the basic definitions. 
[2] I adopt "Einstein notation" [coends, Def. 5.3]: superscripts are contravariant and subscripts are covariant components of functors. Co-ending is always on repeated indices, one covariant and one contravariant. There's no particular reason for it, but the fact that equations are cleaner and do not flood the page.
