The filter cartridge I have slotted at the back. I think carbon is in side it, looks that way to me. I have to buy a sponge for my filter.
I don't want to confuse you as when you have too many chefs in the kitchen, the meal is not always so great.
So I'll address these:
Look at the fish. If the fish do not appear to be struggling to swim, do not change whatever setting you have the filter on.
Look at the water surface. If there is a good rippling effect, that is going to help with oxygenation. If there isn't, if you can, raise the water flow until you have and the fish are not struggling to swim in the tank. The water line appears to be the correct height so that there is some splashing coming from the filter.
While Aiken is correct that sponges are better filtering material than the cartridges that come with the HOB filters, not all HOB filters are able to house large enough sponges to accommodate the load in the tank so you don't always have the option of replacing the cartridges.
Look online for sponge cartridges that list your filter brand and model to see if this is an option for you to replace your current cartridges. If it is and you currently have the 2 cartridges, you only want to exchange one cartridge at a time. Start with replacing the new cartridge leaving the established one for later. Wait a month between exchanging the other cartridge. ( You also have the option of running both filters on the tank with the sponge in the old filter and the established cartridge in the new filter. When the old cartridge is worn out and needs replacing, you replace it with the sponge and discontinue using the smaller filter.)
Aiken is also correct that most of the bacteria and microbes that were in the old filter are going to be in the filter cartridge. Moving that cartridge to the new filter would serve the same purpose as running both filters together waiting for the new cartridge to get some bacterial growth. Doing this is the same as "seeding" a new tank using old material. The remedy for any imbalance or loss would be to add some fritz #7 to the tank to help eliminate any ammonia or nitrite rise. You'll want to monitor the ammonia and nitrite levels for at least the first week after the exchange. If anything is going to happen, it should happen within the first few days.
Regarding this other black cartridge: does your new filter have slots for 2 cartridges? If not, you need to make sure that adding that second cartridge is not slowing the water flow. ( You can check this by removing one cartridge and see if the flow increases. ) If your filter has 2 slots, there is no reason not to keep it in there as the more material for the bacteria to adhere to, the better. The carbon in these cartridges does not last a long time ( 30-45 days +/- ) so after that amount of time, making a slit in the top or the cartridge and pouring out the carbon before replacing it in the filter will help keep the water flow going. ( You do want to check the packaging to make sure that the black cartridge is not just a "dummy one' to show the consumer how they go.
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So this is how I would proceed.