Bruised under nail of big toe. Is this fungus?

Probably not. Bruises beneath the toenail are usually caused by something forcefully dropped onto or pressing the toenail. If the nail becomes chronically bruised from shoegear or activity it could lead to enough damage that fungus could begin to grow beneath the nail and cause an infection.
Maybe. If your saying its a bruise then you already diagnosed it as a hematoma not fungus. Fungus can result from trauma to the toe. If its been there for a while fungus might be under the nail with dried blood.

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