Same thing with the carbon filter, Some filters are meant to only filter air efficently one way through the white prefilter and then through the carbon exhausting the filtered air in the center. So if your filter is not specified to be bidirectional, it should be placed inside the grow room/tent and your 440cfm votex fan should be connected to it pulling air through the filter not pushing it through. The vortex fan can be placed either inside or outside the grow room/tent for weight and space reasons. Has long as it's the fan exhausting the air, not the filter your doing the right thing ( if it's bidirectional it does not matter).
P.S your soil looks questionable , you can never go wrong using old fashion gardening store Promix (always wet promix before using it so there's no dry pockets in it), you should keep that in mind! Bonus it comes ph balanced and usually stays ph balanced for a long time!!!