Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^{w_1-local}: i_1$ and $L_2: C \overset{\rightarrow}{\hookleftarrow} C^{w_2-local}: i_2$. I am wondering when $L_1$ restricts to an adjunction $$L_1: C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} C^{w_1, w_2-local}$$ I think this holds when if the endofunctors $i_1\circ L_1$ and $i_2 \circ L_2$ of $C$ commute.

Question: Is there an easily checkable criterion on $w_1, w_2$ to determine whether these endofunctors commute?

Another reasonable condition is to assume that for any morphism $f_1: X_1 \rightarrow Y_1$ in $w_1$, the objects $X_1, Y_1$ are $w_2$-local. In this case, $f_1: X_1 \rightarrow Y_1$ is a morphism in $C^{w_2-local}$, and hence we can form $(C^{w_2-local})^{w_1-local}$ (which I believe is equivalent to $C^{w_1, w_2-local}$) and produce the reflective localization $$ C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} (C^{w_2-local})^{w_1-local} = C^{w_1, w_2-local} $$ However, I am not sure that this localization is the restriction of $L_1$.

I also suspect that this problem might be a special case of a more general setting with pairs of adjoint functors that are not reflective localizations, i.e. a pair of adjoint functors whose composition endofunctors commute.

** I am thinking about this problem in the $\infty$-categorical setting but an answer in the setting of ordinary categories would also be helpful.

Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$-local}: i_1$ and $L_2: C \overset{\rightarrow}{\hookleftarrow} C^\text{$w_2$-local}: i_2$. I am wondering when $L_1$ restricts to an adjunction $$L_1: C^\text{$w_2$-local} \overset{\rightarrow}{\hookleftarrow} C^\text{$w_1$, $w_2$-local}.$$ I think this holds when if the endofunctors $i_1\circ L_1$ and $i_2 \circ L_2$ of $C$ commute.

Question: Is there an easily checkable criterion on $w_1$, $w_2$ to determine whether these endofunctors commute?

Another reasonable condition is to assume that for any morphism $f_1: X_1 \rightarrow Y_1$ in $w_1$, the objects $X_1$, $Y_1$ are $w_2$-local. In this case, $f_1: X_1 \rightarrow Y_1$ is a morphism in $C^\text{$w_2$-local}$, and hence we can form $(C^\text{$w_2$-local})^\text{$w_1$-local}$ (which I believe is equivalent to $C^\text{$w_1$, $w_2$-local}$) and produce the reflective localization $$ C^\text{$w_2$-local} \overset{\rightarrow}{\hookleftarrow} (C^\text{$w_2$-local})^\text{$w_1$-local} = C^\text{$w_1$, $w_2$-local}. $$ However, I am not sure that this localization is the restriction of $L_1$.

I also suspect that this problem might be a special case of a more general setting with pairs of adjoint functors that are not reflective localizations, i.e. a pair of adjoint functors whose composition endofunctors commute.

** I am thinking about this problem in the $\infty$-categorical setting but an answer in the setting of ordinary categories would also be helpful.

Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^{w_1-local}: i_1$ and $L_2: C \overset{\rightarrow}{\hookleftarrow} C^{w_2-local}: i_2$. I am wondering when $L_1$ restricts to an adjunction $$L_1: C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} C^{w_1, w_2-local}$$

One reasonable condition is to assume that for any morphism $f_1: X_1 \rightarrow Y_1$ in $w_1$, the objects $X_1, Y_1$ are $w_2$-local. In this case, $f_1: X_1 \rightarrow Y_1$ is a morphism in $C^{w_2-local}$, and hence we can form $(C^{w_2-local})^{w_1-local}$ (which I believe is equivalent to $C^{w_1, w_2-local}$) and produce the reflective localization $$ C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} (C^{w_2-local})^{w_1-local} = C^{w_1, w_2-local} $$ However, I am not sure that this localization is the restriction of $L_1$.

I also suspect that this problem might be a special case of a more general setting with pairs of adjoint functors that are not reflective localizations.

** I am thinking about this problem in the $\infty$-categorical setting but an answer in the setting of ordinary categories would also be helpful.

Suppose there are two classes of morphisms $w_1, w_2$ in $C$ and two two reflective localizations $L_1: C \overset{\rightarrow}{\hookleftarrow} C^{w_1-local}: i_1$ and $L_2: C \overset{\rightarrow}{\hookleftarrow} C^{w_2-local}: i_2$. I am wondering when $L_1$ restricts to an adjunction $$L_1: C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} C^{w_1, w_2-local}$$ I think this holds when if the endofunctors $i_1\circ L_1$ and $i_2 \circ L_2$ of $C$ commute.

Question: Is there an easily checkable criterion on $w_1, w_2$ to determine whether these endofunctors commute?

Another reasonable condition is to assume that for any morphism $f_1: X_1 \rightarrow Y_1$ in $w_1$, the objects $X_1, Y_1$ are $w_2$-local. In this case, $f_1: X_1 \rightarrow Y_1$ is a morphism in $C^{w_2-local}$, and hence we can form $(C^{w_2-local})^{w_1-local}$ (which I believe is equivalent to $C^{w_1, w_2-local}$) and produce the reflective localization $$ C^{w_2-local} \overset{\rightarrow}{\hookleftarrow} (C^{w_2-local})^{w_1-local} = C^{w_1, w_2-local} $$ However, I am not sure that this localization is the restriction of $L_1$.

I also suspect that this problem might be a special case of a more general setting with pairs of adjoint functors that are not reflective localizations, i.e. a pair of adjoint functors whose composition endofunctors commute.

** I am thinking about this problem in the $\infty$-categorical setting but an answer in the setting of ordinary categories would also be helpful.